This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present techniques. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present techniques. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
Hydrocarbons are widely used for fuels and chemical feedstocks. Hydrocarbons are generally found in subsurface rock formations that are generally termed reservoirs. Removing hydrocarbons from the reservoirs depends on numerous physical properties of the rock formations, such as the permeability of the rock containing the hydrocarbons, the ability of the hydrocarbons to flow through the rock formations, and the proportion of hydrocarbons present, among others.
Often, mathematical models termed “reservoir simulation models” are used to simulate hydrocarbon reservoirs for locating hydrocarbons and optimizing the production of the hydrocarbons. A reservoir simulator models the flow of a multiphase fluid through a heterogeneous porous media, using an iterative, time-stepping process where a particular hydrocarbon production strategy is optimized. Most reservoir simulation models assume linear potential flow. Darcy's law may be used to describe the linear relationship between potential gradients and flow velocity. In some regions of the reservoir, non-Darcy flow models such as Forchheimer flow, which describe a non-linear relationship between potential gradient and fluid velocity, may be used. In general, however, these models were developed assuming single-phase flow. Therefore, reservoir simulators have extended those models to multiphase flow assuming that each phase may be handled separately and or coupled via capillary pressure effects.
Once the governing equations are defined, equations on a simulation grid are discretized. State variables are then updated through time according to the boundary conditions. The accuracy of the solution depends on the assumptions inherent in the discretization method and the grid on which it is applied. For example, a simple two-point flux approximation (TPFA) in conjunction with a finite difference approach assumes that the fluid velocity is a function of potentials of only two points. This is valid if the grid is orthogonal and permeability is isotropic in the region in question. If permeability is not isotropic and/or the grid is not orthogonal, this TPFA is incorrect and the fluid velocity will be inaccurate. Alternative multi-point flux approximations (MPFA) or different discretization methods, such as Finite Element Methods, have been applied to address this problem. Such methods currently suffer from their inability to resolve the problem on complex geometries in a computationally efficient manner.
Properties for reservoir simulation models, such as permeability or porosity, are often highly heterogeneous across a reservoir. The variation may be at all length scales from the smallest to the largest scales that can be comparable to the reservoir size. Disregarding the heterogeneity can often lead to inaccurate results. However, computer simulations that use a very fine grid discretization to capture the heterogeneity are computationally very expensive.
Accordingly, the simulation grid is often relatively coarse. As a consequence, each grid cell represents a large volume (e.g. 100 meters to kilometers on each side of a 3D grid cell). However, physical properties such as rock permeability vary quite significantly over that volume. Most modern simulators start with a fine grid representation of the data and use some version of flow-based scale-up to calculate an effective permeability over the coarse grid volume. However, relative permeability, which is a function of saturation, may change dramatically over the volume of the coarse grid when simulated using a fine grid model. This is handled by both scaling up the absolute permeability and assuming that relative permeability scales uniformly in the volume of the coarse grid cell, or by the use of dynamic pseudo functions for each coarse grid cell block. As currently used, pseudo functions do not provide the reusability and flexibility needed to attain their full potential. For example, a change in boundary conditions (moving a well) requires regeneration of the pseudo functions.
In some cases, a dual permeability simulation model may be used to improve scale-up accuracy. Dual permeability simulation models use methods conceptually similar to the use of pseudo functions in order to generate two-level effective permeabilities and matrix-fracture transfer functions. Furthermore, effects such as hysteresis, where relative permeability is not only a function of saturation, but also direction in which saturation is changing, are treated as special cases. In other words, a property such as phase permeability is a scale and time dependent property that is difficult to scale-up accurately and with a simple model.
A method of using a neural network to determine an optimal placement of wells in a reservoir is described in “Applying Soft Computing Methods to Improve the Computational Tractability of a Subsurface Simulation-Optimization Problem,” by Virginia M. Johnson & Leah L. Rogers, 29 JOURNAL OF PETROLEUM SCIENCE AND ENGINEERING 2001.153-175 (2001). Using a standard industry reservoir simulator, a knowledge base of 550 simulations sampling different combinations of 25 potential injection locations was created. Neural networks were trained from a representative sample of simulations, which forms a re-useable knowledge base of information for addressing many different management questions. Artificial neural networks (ANNs) were trained to predict peak injection volumes and volumes of produced oil and gas at three and seven years after the commencement of injection. The rapid estimates of these quantities provided by the ANNs were fed into net profit calculations, which in turn were used by a genetic algorithm (GA) to evaluate the effectiveness of different well-field scenarios.
Methods of using different types of neural networks as proxies to a reservoir simulation are described in “Use of Neuro-Simulation techniques as proxies to reservoir simulator: Application in production history matching,” by Paulo Camargo Silva, et al., JOURNAL OF PETROLEUM SCIENCE AND ENGINEERING 57 273-280 (2007). In this article, different types of Neural Networks were used as proxies to a reservoir simulator. The Neural Networks were applied in a study of production history matching for a synthetic case and a real case. A reservoir simulator was used to generate training sets for the Neural Networks. And for these cases, the authors were able to reproduce the narrowly modeled behavior response via the ANN.
Methods to provide an improved and faster history matching with a nonlinear proxy are described in “Improved and More-Rapid History Matching with a nonlinear Proxy and Global Optimization,” by A. S. Cullick, et al., SPE 101933, SOCIETY OF PETROLEUM ENGINEERS (2008). A comprehensive, nonlinear proxy neural network is trained with a relatively small set of numerical simulations that were defined through a design of experiments (DOE). The neural network is used to characterize parameter sensitivities to reservoir parameters and to generate solution sets of the parameters that match history. The solution sets generated by the neural network can be validated with the simulator or used as initial solutions for a full optimization.
Additional background information can be found in “Smooth Function Approximation Using Neural Networks,” by Silvia Ferrari & Robert F. Stengel, IEEE TRANSACTIONS ON NEURAL NETWORKS, Vol. 16, No. 1 (2005); and “Introducing Predictive Analytics: Opportunities,” by Paul Stone, SPE 106865, Society of Petroleum Engineers (2007). None of the techniques described above provide a fast and accurate method of using a machine learning techniques to compute a machine learning based solution surrogate for performing a reservoir simulation.